Recent years have witnessed a growing importance of lithium secondary batteries as power sources installed in vehicles, and as power sources in personal computers, cell phones and the like. Lithium ion secondary batteries, in particular, are lightweight and afford high energy densities, and are thus expected to become preferred high-output power sources installed in vehicles. Typical examples of positive electrode active materials used in lithium secondary batteries include complex oxides that comprise lithium and a transition metal element. For instance, there is preferably used a complex oxide in the form of a lithium complex oxide that contains at least nickel (Ni) as the transition metal element (nickel-containing lithium complex oxide), and that has a layered structure. Prior art documents relating to lithium secondary batteries include Patent documents 1 to 4 below.    Patent document 1: Japanese Patent Application Publication No. 2009-140787    Patent document 2: Japanese Patent Application Publication No. 2008-53054    Patent document 3: Japanese Patent Application Publication No. 2007-299668    Patent document 4: Japanese Patent Application Publication No. 2007-273448